几类非全局Lipschitz条件下随机延迟微分方程的数值分析

近年来，在金融、经济等学科中涌现出越来越多不满足全局李普希茨条件甚至系数不连续的随机延迟微分方程模型，此类方程解的性质及其数值分析亟待解决。本项目拟讨论几类系数较弱条件下随机延迟微分方程的数值分析。首先，对漂移项超线性增长且漂移项和扩散项满足局部李普希茨条件下的随机延迟微分方程，讨论theta-EM算法解的存在唯一性、矩有限性和强收敛阶数。其次，给出满足单调性和全局单边李普希茨条件以及单调性和局部单边李普希茨条件下的随机延迟微分方程theta-EM算法解的性质。然后，考察漂移项系数不连续时一维随机延迟微分方程强解的存在唯一性，EM算法的存在唯一性、矩有限性和强收敛性等。最后，将漂移项系数不连续的一维随机延迟微分方程推广至高维情形，讨论解的性质及其数值模拟，并进一步讨论漂移和扩散系数均不连续的方程解和数值解的相关性质。本项目的研究可望为充实随机微分方程数值解理论和推动其它学科的发展做出贡献。

Recently, more and more applications related to stochastic differential delay equations with non-global Lipschitz conditions or even with discontinuous coefficients have emerged in subjects such as finance and economics. Properties of solutions and numerical analysis to those kind of equations need to be known as soon as possible. In this project, we are going to discuss the existence and uniqueness, p-th moment finiteness and strong convergence of numerical schemes to stochastic differential delay equations under non-global Lipschitz conditions or with discontinuous coefficients. First, we discuss the existence and uniqueness, p-th moment finiteness and strong convergence rate of theta-EM scheme to stochastic differential delay equations with the drift coefficient growing superlinearly and satisfying the local Lipschitz condition. Next, to stochastic differential delay equations with global one-sided Lipschitz and monotone condition, discuss the strong convergence rate of the theta-EM scheme, then, weaker the global one-sided Lipschitz condition to the local one-sided Lipschitz condition, analysis the strong convergence of the theta-EM scheme. Then, for scalar stochastic differential delay equations with discontinuous drift coefficient, discuss the existence and uniqueness of the strong solution, strong convergence rate of the EM scheme. At last, discuss the properties of the solution to multidimensional stochastic differential delay equations with discontinuous drift coefficient, then analysis the strong convergence of the EM solution to the exact solution. Furthermore, discuss properties of stochastic differential delay equations with discontinuous drift coefficient and discontinuous diffusion coefficient. This project will not only enrich the theory of numerical analysis to stochastic differential equations, but also promote the development of other subjects.